Cartan sub-algebra

In mathematics , especially in the theory of Lie algebras , Cartan subalgebras are used, among other things, in the classification of semi-simple Lie algebras and in the theory of symmetric spaces . The rank of a Lie algebra (or its associated Lie group ) is defined as the dimension of the Cartan subalgebra. An example of a Cartan sub-algebra is the algebra of diagonal matrices.

definition

Let it be a Lie algebra. A subalgebra is a Cartan subalgebra if it is nilpotent and self-normalizing , that is, if ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {a}}\subset {\mathfrak {g}}}$

• ${\displaystyle \underbrace {[{\mathfrak {a}},[{\mathfrak {a}},[\dotsb ,[{\mathfrak {a}},{\mathfrak {a}}} _{n}]\dotsb ]]]=0}$  for one and ${\displaystyle n\in \mathbb {N} }$
• ${\displaystyle \forall Y\not \in {\mathfrak {a}}\ \exists X\in {\mathfrak {a}}:\ \left[X,Y\right]\not \in {\mathfrak {a}}}$

applies.

Examples

A Cartan subalgebra of

${\displaystyle {\mathfrak {g}}={\mathfrak {sl}}(n,\mathbb {C} )=\left\{A\in \mathrm {Mat} (n,\mathbb {C} ):\mathrm {Spur} (A)=0\right\}}$

is the algebra of the diagonal matrices

${\displaystyle {\mathfrak {a}}_{0}=\left\{\mathrm {diag} (\lambda _{1},\ldots ,\lambda _{n}):\lambda _{1}+\ldots +\lambda _{n}=0\right\}}$.

Every Cartan sub-algebra is too conjugate . ${\displaystyle {\mathfrak {a}}\subset {\mathfrak {sl}}(n,\mathbb {C} )}$ ${\displaystyle {\mathfrak {a}}_{0}}$

In contrast, Cartan has two unconjugated subalgebras, viz ${\displaystyle {\mathfrak {sl}}(2,\mathbb {R} )}$

${\displaystyle {\mathfrak {a}}_{1}=\mathbb {R} \left({\begin{array}{cc}1&0\\0&-1\end{array}}\right)}$

and

${\displaystyle {\mathfrak {a}}_{2}=\mathbb {R} \left({\begin{array}{cc}0&1\\1&0\end{array}}\right)}$.

Existence and uniqueness

A finite-dimensional Lie algebra over an infinite field always has a Cartan subalgebra.

For a finite-dimensional Lie algebra over a field with characteristic it holds that all Cartan subalgebras have the same dimension. ${\displaystyle 0}$

For a finite-dimensional Lie algebra over an algebraically closed field , all Cartan subalgebras are conjugated to one another , namely under the group that is generated by the automorphisms (for in Lie algebra and nilpotent). ${\displaystyle \exp(\mathrm {ad} (X))}$ ${\displaystyle X}$ ${\displaystyle \mathrm {ad} (X)}$

properties

If a semi-simple Lie algebra is over an algebraically closed field , then every Cartan sub -algebra is Abelian and the restriction of the adjoint representation to is simultaneously diagonalizable with as eigenspace to weight . That is, there is a decomposition ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {a}}\subset {\mathfrak {g}}}$ ${\displaystyle \mathrm {ad} \colon {\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}})}$ ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle 0}$

${\displaystyle {\mathfrak {g}}={\mathfrak {a}}\oplus \bigoplus _{\alpha \in {\mathfrak {a}}^{*}}{\mathfrak {g}}_{\alpha }}$

With

${\displaystyle \mathrm {ad} (X)(Y)=\left[X,Y\right]=\alpha (X)Y\quad \forall X\in {\mathfrak {a}},Y\in {\mathfrak {g}}_{\alpha }}$

and

${\displaystyle {\mathfrak {g}}_{\alpha }\not =0\Longrightarrow \alpha (X)\not =0\quad \forall X\in {\mathfrak {a}}}$.

For example

${\displaystyle {\mathfrak {g}}={\mathfrak {sl}}(n,\mathbb {C} )=\left\{A\in \mathrm {Mat} (n,\mathbb {C} ):\mathrm {Spur} (A)=0\right\},}$

${\displaystyle {\mathfrak {a}}=\left\{\mathrm {diag} (\lambda _{1},\ldots ,\lambda _{n}):\lambda _{1}+\ldots +\lambda _{n}=0\right\}}$

is when the elementary matrix is designated with entry at that point and entries otherwise ${\displaystyle e_{ij}}$ ${\displaystyle 1}$ ${\displaystyle (i,j)}$ ${\displaystyle 0}$

${\displaystyle {\mathfrak {g}}={\mathfrak {a}}\oplus \bigoplus _{i\not =j}\mathbb {C} e_{ij}={\mathfrak {a}}\oplus \bigoplus _{\alpha \in {\mathfrak {a}}^{*}}{\mathfrak {g}}_{\alpha }}$

with for ${\displaystyle \mathbb {C} e_{ij}={\mathfrak {g}}_{\alpha }}$

${\displaystyle \alpha (\lambda _{1},\ldots ,\lambda _{n})=\lambda _{i}-\lambda _{j}}$.

literature

• Élie Cartan : Sur la structure des groupes de transformations finis et continus. Thèse, Paris 1894.
• Anthony W. Knapp: Lie groups beyond an introduction. (Progress in Mathematics, 140). Second edition. Birkhäuser, Boston, MA 2002, ISBN 0-8176-4259-5 .